4.10.9 \(y'(x) (a+b x+y(x))+a-b x-y(x)=0\)

ODE
\[ y'(x) (a+b x+y(x))+a-b x-y(x)=0 \] ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 0.0559698 (sec), leaf count = 49

\[\left \{\left \{y(x)\to \frac {2 a W\left (-e^{\frac {(b+1)^2 x}{2 a}+c_1-1}\right )+a (-b)+a-b (b+1) x}{b+1}\right \}\right \}\]

Maple
cpu = 0.032 (sec), leaf count = 71

\[ \left \{ {\frac {a \left (-1+b \right ) \ln \left (\left (b+1 \right ) y \relax (x ) +{b}^{2}x+ \left (a+x \right ) b-a \right ) - \left (b+1 \right ) \left (a\ln \left (\left (b+1 \right ) \left (bx+y \relax (x ) \right ) +ab-a \right ) +{\it \_C1}\,b-x+{\it \_C1}+y \relax (x ) \right ) }{ \left (b+1 \right ) ^{2}}}=0 \right \} \] Mathematica raw input

DSolve[a - b*x - y[x] + (a + b*x + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (a - a*b - b*(1 + b)*x + 2*a*ProductLog[-E^(-1 + ((1 + b)^2*x)/(2*a) +
 C[1])])/(1 + b)}}

Maple raw input

dsolve((a+b*x+y(x))*diff(y(x),x)+a-b*x-y(x) = 0, y(x),'implicit')

Maple raw output

(a*(-1+b)*ln((b+1)*y(x)+b^2*x+(a+x)*b-a)-(b+1)*(a*ln((b+1)*(b*x+y(x))+a*b-a)+_C1
*b-x+_C1+y(x)))/(b+1)^2 = 0